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17. Use the theory of residue to evaluate the following definite integrals r2 (a) /" (b) / do 2+ cos 0 ' cos 30d0 5 – 4 cos 0' - cos 20d0 1 — 2а сos @ + a2 (-1 < a < 1), (c) where (4) / sin?" Odo where n = 1,2, ....

17. Use the theory of residue to evaluate the following definite integrals r2 (a) /" (b) / do 2+ cos 0 ' cos 30d0 5 – 4 cos 0' - cos 20d0 1 — 2а сos @ + a2 (-1 < a < 1), (c) where (4) / sin?" Odo where n = 1,2, ....

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve 17(b) and 17(c) 

**Exercise 17** Use the theory of residue to evaluate the following definite integrals: (a) \( \int_{0}^{2\pi} \frac{d\theta}{2 + \cos \theta}, \) (b) \( \int_{0}^{2\pi} \frac{\cos 3\theta \,d\theta}{5 - 4 \cos \theta}, \) (c) \( \int_{0}^{\pi} \frac{\cos 2\theta \,d\theta}{1 - 2a \cos \theta + a^2} \) where \((-1 < a < 1),\) (d) \( \int_{0}^{\pi} \sin^{2n} \theta \, d\theta \) where \(n = 1, 2, \ldots\). **Answers:** (a) \( \frac{2\pi}{\sqrt{3}}, \) (b) \( \frac{\pi}{12}, \) (c) \( \frac{\pi a^2}{1 - a^2}, \) (d) \( \frac{\pi \, (2n)!}{2^{2n}(n!)^2}. \)
Transcribed Image Text:**Exercise 17** Use the theory of residue to evaluate the following definite integrals: (a) \( \int_{0}^{2\pi} \frac{d\theta}{2 + \cos \theta}, \) (b) \( \int_{0}^{2\pi} \frac{\cos 3\theta \,d\theta}{5 - 4 \cos \theta}, \) (c) \( \int_{0}^{\pi} \frac{\cos 2\theta \,d\theta}{1 - 2a \cos \theta + a^2} \) where \((-1 < a < 1),\) (d) \( \int_{0}^{\pi} \sin^{2n} \theta \, d\theta \) where \(n = 1, 2, \ldots\). **Answers:** (a) \( \frac{2\pi}{\sqrt{3}}, \) (b) \( \frac{\pi}{12}, \) (c) \( \frac{\pi a^2}{1 - a^2}, \) (d) \( \frac{\pi \, (2n)!}{2^{2n}(n!)^2}. \)
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